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====== Higher hyperoctahedral series ======

The **higher hyperoctahedral series** is a family $(H_N^{[s]})_{N\in \N,\,s\in\{3,\ldots,\infty\}}$ of [[compact matrix quantum group|compact matrix quantum groups]] introduced by Banica, Curran and Speicher in [(:ref:BanCuSp10)]. Each $H_N^{[s]}$ interpolates the quantum group $H_N^{(s)}$  of the [[hyperoctahedral series]] with parameter $s$ and the [[free hyperoctahedral quantum group]] $H_N^{+}$, both of the corresponding dimension $N$.

===== Definition =====
Given $N\in \N$ and $s\in\N\cup\{\infty\}$ with $s\geq 3$, the **quantum group**  $H_N^{[s]}$ **of the hyperoctahedral series with parameter** $s$ **for dimension** $N$ is the [[compact matrix quantum group]] $(C(H_N^{[s]}),u)$ where $u=(u_{i,j})_{i,j=1}^N$ organizes the generators $\{u_{i,j}\}_{i,j=1}^N$ of the (unital) [[wp>Universal_C*-algebra|universal C*-algebra]] 
$$C(H_N^{[s]})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,u=\overline u, \, uu^t=u^tu=I_N\otimes 1\,$$
$${\color{white}C(H_N^{(s)})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall_{i,j,k=1}^N: i\neq j\Rightarrow u_{i,k}u_{j,k}=u_{k,i}u_{k,j}=0,$$
$${\color{white}C(H_N^{(s)})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}\forall_{i,j,k,l=1}^N: u_{i,j}^2u_{k,l}=u_{k,l}u_{i,j}^2,$$
$${\color{white}C(H_N^{(s)})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}s<\infty \Rightarrow \forall a,b\in \{u_{i,j}\}_{i,j=1}^n: $$
$${\color{white}C(H_N^{(s)})\colon\hspace{-0.66em}= C^\ast_1\big\langle\{u_{i,j}\}_{i,j=1}^N\big\,\vert \,}(s\text{ odd} \Rightarrow (ab)^{\frac{s-1}{2}}a=(ba)^{\frac{s-1}{2}}b),\, (s\text{ even}\Rightarrow (ab)^{\frac{s}{2}}=(ba)^{\frac{s}{2}})\big\rangle,$$
where $\overline u=(u^\ast_{i,j})_{i,j=1}^N$ is the complex conjugate of $u$ and $u^t=(u_{j,i})_{i,j=1}^N$ the transpose, where $I_N$ is the identity $N\!\times \!N$-matrix and where $1$ is the unit of the universal $C^\ast$-algebra.

If $s<\infty$, the definition can also be expressed by saying that the fundamental corpresentation matrix $u$ of $H_N^{[s]}$ is **cubic** and satisfies the $s$**-mixing relations**. In particular, $u$ satisfies the **ultracubic relations**, which is to say $u_{i,j}u_{l,m}u_{i,k}=u_{k,i}u_{l,m}u_{k,j}=0$ for all $i,j,k,l,m=1,\ldots,N$.

Had one allowed $s=2$ in the definition, one would have obtained the [[hyperoctahedral group|hyperoctahedral group]] $H_N^{[2]}\colon\hspace{-0.66em}=H_N$. 

The quantum groups of the higher hyperoctahedral series are [[group-theoretical_hyperoctahedral_easy_orthogonal_quantum_groups|group-theoretical hyperoctahedral orthogonal easy quantum groups]] and can therefore be written as a [[semi-direct product]] with its [[diagonal subgroup of a compact matrix quantum group|diagonal subgroup]] [(:ref:RaWe15)]: 
$$C(H_N^{[s]})\cong C^\ast\langle \{a_i\}_{i=1}^n \,\vert\, \forall_{i,j=1}^n: a_i^2=1,\, s<\infty\Rightarrow (a_ia_j)^s=1\rangle\bowtie C(S_N)$$
for all $N\in \N$ and $s\in\N\cup\{\infty\}$ with $3\leq s$, where $C(S_N)$ denotes the continuous functions over the symmetric group of dimension $N$ (considered as the subgroup of $\mathrm{GL}(\C,N)$ given by all [[wp>permutation matrices]]).

===== Basic Properties =====

The fundamental corepresentation matrix $u$ of $H_N^{[s]}$ is in particular //orthogonal//. Hence, $H_N^{[s]}$ is a compact quantum subgroup of the [[free orthogonal quantum group]] $O_N^+$.

Moreover, $u$ is also //cubic// especially, implying that $H_N^{[s]}$ is a compact quantum subgroup of the [[free hyperoctahedral quantum group]] $H_N^{+}$, the free counterpart of the hyperoctahedral group $H_N$.

If $I$ denotes the closed two-sided ideal of $C(H_N^{[s]})$ generated by the relations $u_{i,j}u_{k,l}=u_{k,l}u_{i,j}$ for any $i,j,k,l=1,\ldots, N$, then $C(H_N^{[s]})/I$ is isomorphic to the $C^\ast$-algebra $C(H_N)$ of continuous functions on the [[hyperoctahedral group]] $H_N$, the subgroup  of $\mathrm{GL}(N,\C)$ given by orthogonal matrices with integer entries. Hence, $H_N^{[s]}$ is a compact quantum supergroup of $H_N$.

Similarly, if $J$ is the closed two-sided ideal of $C(H_N^{[s]})$ generated by the relations $acb=bca$ for any $a,b,c\in \{u_{i,j}\}_{i,j=1}^N$, then $C(H_N^{[s]})/J$ is isomorphic to the $C^\ast$-algebra $C(H_N^\ast)$ of the [[half-liberated hyperoctahedral quantum group]] $H_N^\ast$. Hence, $H_N^{[s]}$ is a compact quantum supergroup of $H_N^\ast$.


For every $s\in \N$ with $s\geq 3$ the quantum groups $(H_N^{[s]})_{N\in \N}$ of the higher hyperoctahedral series with parameter $s$ are an [[easy_quantum_group|easy]] family of compact matrix quantum groups, i.e., the intertwiner spaces of their corepresentation categories are induced by a [[category of partitions]]. More precisely, it is a [[group-theoretical hyperoctahedral categories of partitions|group-theoretical hyperoctahedral category of partitions]] that induces the corepresentation categories of $(H_N^{[s]})_{N\in \N}$. Canonically, if $s<\infty$, it is generated by $h_s$  [(:ref:RaWe14)], the partition whose [[partition#word_representation|word representation]] is given by $(\mathsf{ab})^s$. See also [[categories of the higher hyperoctahedral series]]. The corepresentation categories of $(H_N^{[\infty]})_{N\in\N}$ are induced by $\Paabaab$.


===== Representation theory =====


===== Cohomology =====


===== Related quantum groups =====

===== References =====

[( :ref:BanSp09 >>
author:  Banica, Teodor and Speicher, Roland
title:   Liberation of orthogonal Lie groups
year:    2009
journal: Advances in Mathematics
volume:  222
issue:   4
pages:   1461--150
url:     https://doi.org/10.1016/j.aim.2009.06.009
archivePrefix: arXiv
eprint   :0808.2628
)]


[( :ref:Web12 >>
author:  Weber, Moritz
title:   On the classification of easy quantum groups
year:    2013
journal: Advances in Mathematics
volume:  245
pages:   500--533
url:     https://doi.org/10.1016/j.aim.2013.06.019
archivePrefix: arXiv
eprint   :1201.4723v2
)]

[( :ref:BanCuSp10 >>
author:  Banica, Teodor and Curran, Stephen and Speicher, Roland
title:   Classification results for easy quantum groups
year:    2010
journal: Pacific Journal of Mathematics
volume:  247
issue:   1
pages:   1-26
url:     https://doi.org/10.2140/pjm.2010.247.1
archivePrefix: arXiv
eprint   :0906.3890v1
)]

[( :ref:RaWe15 >>
author:  Raum, Sven and Weber, Moritz 
title:   Easy quantum groups and quantum subgroups of a semi-direct product quantum group
year:    2015
journal: Journal of Noncommutative Geometry
volume:  9
issue:   4
pages:   1261--1293
url:     https://doi.org/10.4171/JNCG/223 
archivePrefix: arXiv
eprint   :1311.7630v2
)]

[( :ref:RaWe14 >>
author:  Raum, Sven and Weber, Moritz 
title:   The combinatorics of an algebraic class of easy quantum groups
year:    2014
journal: Infinite Dimensional Analysis, Quantum Probability and related topics
volume:  17
issue:   3
url:     https://doi.org/10.1142/S0219025714500167
archivePrefix: arXiv
eprint   :1312.1497v1
)]
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